Problem 6

Show that there exists a set

A

of positive integers with the following property: for any infinite set

S

of primes, there exist two positive integers

m\in A

and

n\notin A

, each of which is a product of

k

distinct elements of

S

for some

k\ge2

Step 2 of 4: Index an arbitrary infinite set

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In plain words

The first prime in $S$ supplies a legal block length $k\ge2$ .

S=\{s_1<s_2<\cdots\},\quad k=s_1

Detailed analysis

Write the infinite set as $S=\{s_1<s_2<\cdots\}$ and set $k=s_1$ . Since $s_1$ is a prime, $k\ge2$ , and infinitely many elements remain after the first $k+1$ .